In subband coding a wideband signal in the frequency domain is subjected to a spectrum analysis procedure to apportion its energy content among a plurality of subbands, which are individually coded. The coding procedures usually involve decimation. The coding procedures may include different quantizing thresholds for the various subbands. The coding procedures may involve statistical coding. The codes are transmitted and are decoded to recover the spectrum analyses. Decimated subband information is expanded through interpolation and combined to synthesize a replica of the original wideband signal.
A type of filtering called quadrature mirror filtering is well adapted for use in subband coding and in other signal processing that involves the spectral analysis of a wideband signal, operation on the subspectra, and the synthesis of a wideband signal from the operated on subspectra. A quadrature-mirror-filter (QMF) bank of filters divides a band of n-dimensional frequencies into 2.sup.n subbands with cross-overs in response between subbands at exactly one-half Nyquist frequency. Each subband is subsampled 2:1 in each of the n dimensions as compared to the band from which it was separated by the QMF. The number n is a positive integer. The number n is one, when simple temporal frequencies or linear spatial frequencies are being filtered. The number n is two when transversal filtering of temporal frequencies is involved or when there is direct two-dimensional spatial frequency filtering involved. The quadrature mirror filters are complementary in their responses, each exhibiting 6 dB attenuation at half Nyquist frequency in a dimension, so that summing their responses (after resampling them to the density of samples in the band supplied to the QMF bank) provides an all-pass network. The low-pass subband and high-pass subband portions of filter response H(z) and H(-z) in each of the n dimensions have frequency spectra that are complements of each other and sum to a flat response.
A primer for QMF bank design is found in Subchapter 7.7 "Filter Banks Based On Cascaded Realizations and Tree Structures" on pages 376-395 of the book MULTIRATE DIGITAL SIGNAL PROCESSING by R. E. Crochiere and Lawrence R. Rabiner published by Prentice-Hall Inc., Englewood Cliff N.J. 07632. As pointed out on page 383 of this book the low-pass filter kernels and high-pass filter kernels, which are used in conventional quadrature mirror filters of symmetric finite impulse response (FIR) type, invariably have an even number of taps in order to be useful. The low-pass filters exhibit cosine (or even) symmetry, of course, but the high-pass filters exhibit sine (or odd) symmetry. Subsampling of each filter response is done in phase with that of the other. Crochiere and Rabiner indicate that a set of conditions for QMF bank design alternative to conventional practice do exist, however, which allow the use of filter kernels that have an odd number of taps.
C. R. Galand and H. J. Nussbaumer in their paper "New Quadrature Mirror Filter Structures" appearing on pages 522-531 of IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, Vol. ASSP-32, No. 3 June 1984 include a section "IV. Odd Quadrature Mirror Filters." They describe modifications of the conventional QMF banks, introducing a one-sample delay in the high-pass filter of the synthesizing QMF bank used to separate the input signal into subbands, and introducing a one-sample delay in the low-pass filter of the analyzing QMF bank used to reconstruct the input signal from the subbands, which modifications accommodate the use of filter kernels with odd numbers of taps. Decimations of the low-pass and high-pass filter responses are done in phase.
G. Wackersreuther in his paper "On The Design Of Filters For Ideal QMF and Polyphase Filter Banks" appearing in pp. 123-130 of ARCHIV FOR ELECTRONIK UND UBERTRAGUNSTECHNIK, Vol. 39, No. 2, February 1985, describes an alternative configuration to that suggested by Galand and Nussbaumer, in which alternative configuration decimations of the low-pass and high-pass filter responses are done out-of-phase, thus eliminating the need for one-sample delays to accommodate the use of filter kernels with odd numbers of taps.